Statistical Science
2014, Vol. 29, No. 1, 106–112
DOI: 10.1214/13-STS417
© Institute of Mathematical Statistics, 2014
The Two-Piece Normal, Binormal, or
Double Gaussian Distribution: Its Origin
and Rediscoveries
Kenneth F. Wallis
Abstract. This paper traces the history of the two-piece normal distribution from its origin in the posthumous Kollektivmasslehre (1897) of Gustav
Theodor Fechner to its rediscoveries and generalisations. The denial of Fechner’s originality by Karl Pearson, reiterated a century later by Oscar Sheynin,
is shown to be without foundation.
Key words and phrases: Gustav Theodor Fechner, Gottlob Friedrich Lipps,
Francis Ysidro Edgeworth, Karl Pearson, Francis Galton, Oscar Sheynin.
translates as the “two-sided Gaussian law,” and refers
to it as “the Fechner distribution” (page 378). However Fechner’s claim to originality had been disputed
by Pearson (1905), whose denial of Fechner’s originality has recently been repeated by Sheynin (2004). In
this paper we reappraise the source and nature of the
various claims, and record several rediscoveries of the
distribution and extensions of Fechner’s basic ideas. As
a prelude to the discussion, there follows a brief technical introduction to the distribution.
A random variable X has a two-piece normal distribution with parameters μ, σ1 and σ2 if it has probability density function (PDF)
1. INTRODUCTION
The two-piece normal distribution came to public
attention in the late 1990s, when the Bank of England and the Sveriges Riksbank began to publish probability forecasts of future inflation, using this distribution to represent the possibility that the balance of
risks around the central forecast might not be symmetric. The forecast probabilities that future inflation
would fall in given intervals could be conveniently calculated by scaling standard normal probabilities, and
the resulting density forecasts were visualised in the
famous forecast fan charts. In both cases the authors
of the supporting technical documentation (Britton,
Fisher and Whitley, 1998; Blix and Sellin, 1998) refer readers to Johnson, Kotz and Balakrishnan (1994)
for discussion of the distribution. These last authors
state (page 173) that “the distribution was originally
introduced by Gibbons and Mylroie (1973),” a reference that post-dates the first edition of Distributions
in Statistics (Johnson and Kotz, 1970), in which the
two-piece normal distribution made no appearance, under this or any other name. On the contrary, the distribution was originally introduced in Fechner’s Kollektivmasslehre (1897) as the Zweispaltiges or Zweiseitige Gauss’sche Gesetz. In his monumental history of
statistics, Hald (1998) prefers the latter name, which
A exp −(x − μ)2 /2σ12 ,
x ≤ μ,
(1) f (x) =
2
2
A exp −(x − μ) /2σ2 ,
x ≥ μ,
√
where A = ( 2π(σ1 + σ2 )/2)−1 . The distribution is
formed by taking the left half of a normal distribution
with parameters (μ, σ1 ) and the right half of a normal
distribution with parameters (μ, σ2 ), and scaling them
to give the common value f (μ) = A at the mode, μ,
as in (1). The scaling factor applied to the left half
of the N(μ, σ1 ) PDF is 2σ1 /(σ1 + σ2 ) while that applied to the right half of N(μ, σ2 ) is 2σ2 /(σ1 + σ2 ),
so the probability mass under the left or right piece is
σ1 /(σ1 + σ2 ) or σ2 /(σ1 + σ2 ), respectively. An example with σ1 < σ2 , in which the two-piece normal distribution is positively skewed, is shown in Figure 1. The
skewness becomes extreme as σ1 → 0 and the distribution collapses to the half-normal distribution, while the
skewness is reduced as σ1 → σ2 , reaching zero when
Kenneth F. Wallis is Emeritus Professor of Econometrics,
Department of Economics, University of Warwick, Coventry
CV4 7AL, United Kingdom (e-mail:
K.F.Wallis@warwick.ac.uk).
106
THE TWO-PIECE NORMAL DISTRIBUTION
F IG . 1. The probability density function of the two-piece normal
distribution. Dashed line: left half of N (μ, σ1 ) and right half of
N (μ, σ2 ) distributions with μ = 2.5 and σ1 < σ2 . Solid line: the
two-piece normal distribution.
σ1 = σ2 and the distribution is again the normal distribution.
The mean and variance of the distribution are
(2)
E(X) = μ +
(3)
2
(σ2 − σ1 ),
π
2
(σ2 − σ1 )2 + σ1 σ2 .
var(X) = 1 −
π
Expressions for the third and fourth moments about the
mean are increasingly complicated and uninformative.
Skewness is more readily interpreted in terms of the ratio of the areas under the two pieces of the PDF, which
is σ1 /σ2 , or a monotone transformation thereof such as
(σ2 − σ1 )/(σ2 + σ1 ), which is the value taken by the
skewness measure of Arnold and Groeneveld (1995).
With only three parameters there is a one-to-one relation between (the absolute value of) skewness and
kurtosis. The conventional moment-based measure of
kurtosis, β2 , ranges from 3 (symmetry) to 3.8692 (the
half-normal extreme asymmetry), hence the distribution is leptokurtic.
Quantiles of the distribution can be conveniently
obtained by scaling the appropriate standard normal
quantiles. For the respective cumulative distribution
functions (CDFs) F (x) and (z) we define quantiles
xp = F −1 (p) and zp = −1 (p). Then in the left piece
of the distribution we have xα = σ1 zβ + μ, where
β = α(σ1 + σ2 )/2σ1 . And in the right piece of the
distribution, defining quantiles with reference to their
upper tail probabilities, we have x1−α = σ2 z1−δ + μ,
where δ = α(σ1 + σ2 )/2σ2 . In particular, with σ1 < σ2 ,
as in Figure 1, the median of the distribution is x0.5 =
σ2 −1 (1 − (σ1 + σ2 )/4σ2 ) + μ. In this case the three
107
central values are ordered mean>median>mode; with
negative skewness this order is reversed.
Although the two-piece normal PDF is continuous
at μ, its first derivative is not and the second derivative
has a break at μ, as first noted by Ranke and Greiner
(1904). This has the disadvantage of making standard
asymptotic likelihood theory inapplicable, nevertheless standard asymptotic results are available by direct
proof for the specific example.
The remainder of this paper is organised as follows.
In Section 2 we revisit the distribution’s origin in Gustav Theodor Fechner’s Kollektivmasslehre, edited by
Gottlob Friedrich Lipps and published in 1897, ten
years after Fechner’s death. In Section 3 we note an
early rediscovery, two years later, by Francis Ysidro
Edgeworth. In Section 4 we turn to the first discussion in the English language of Fechner’s contribution,
in a characteristically long and argumentative article
by Karl Pearson (1905). Pearson derives some properties of “Fechner’s double Gaussian curve,” but asserts
that it is “historically incorrect to attribute [it] to Fechner.” We re-examine Pearson’s evidence in support of
this position, in particular having in mind its reappearance in Oscar Sheynin’s (2004) appraisal of Fechner’s
statistical work. Pearson also argues that “the curve is
not general enough,” especially in comparison with his
family of curves. The overall result was that the Fechner distribution was overlooked for some time, to the
extent that there have been several independent rediscoveries of the distribution in more recent years; these
are noted in Section 5, together with some extensions.
2. THE ORIGINATORS: FECHNER AND LIPPS
Gustav Theodor Fechner (1801–1887) is known as
the founder of psychophysics, the study of the relation
between psychological sensation and physical stimulus, through his 1860 book Elemente der Psychophysik.
Stigler’s (1986, pages 242–254) assessment of this
“landmark” contribution concludes that “at a stroke,
Fechner had created a methodology for a new quantitative psychology.” However, his final work, Kollektivmasslehre, is devoted more generally to the study of
mass phenomena and the search for empirical regularities therein, with examples of frequency distributions
taken from many fields, including aesthetics, anthropology, astronomy, botany, meteorology and zoology.
In his Foreword, Fechner mentions the long gestation
period of the book, and states its main objective as the
establishment of a generalisation of the Gaussian law
of random errors, to overcome its limitations of symmetric probabilities and relatively small positive and
108
K. F. WALLIS
negative deviations from the arithmetic mean. He also
appeals to astronomical and statistical institutes to use
their mechanical calculation powers to produce accurate tables of the Gaussian distribution, which he had
desperately missed during his work on the book. But
the book had not been completed when Fechner died
in November 1887.
The eventual publication of Kollektivmasslehre in
1897 followed extensive work on the incomplete
manuscript by Gottlob Friedrich Lipps (1865–1931).
In his Editor’s Preface, Lipps says that he received
the manuscript in early 1895 and that material he has
worked on is placed in square brackets in the published
work. It is not clear how much unfinished material was
left behind by Fechner or to what extent Lipps had to
guess at Fechner’s intentions. It would appear that the
overall structure of the book had already been set out
by Fechner, since most of the later chapters have early
paragraphs by Fechner, before square-bracketed paragraphs begin to appear. Also, some earlier chapters by
Fechner have forward references to later material that
appears in square brackets. In general, Lipps’ material is more mathematical: he was more of a mathematician than Fechner, who perhaps had set some sections aside for attention later, only to run out of time.
Lipps also has a lighter style: for example, Sheynin
(2004, page 54) complains about some earlier work
that “Fechner’s style is troublesome. Very often his
sentences occupy eight lines, and sometimes much
more—sentences of up to 16 lines are easy to find.”
The same is true of the present work.
The origin of the two-piece normal distribution is
in Chapter 5 of Kollektivmasslehre, titled “The Gaussian law of random deviations and its generalisations.”
Here Fechner uses very little mathematics, postponing more analytical treatment to later chapters. He first
presents a numerical example of the use of the Gaussian distribution to calculate the probability of an observation falling in a given interval. The measure of
location is the arithmetic mean, A, and the measure of
dispersion is the mean absolute deviation, ε (related to
the standard
deviation, in the Gaussian distribution, by
√
ε = σ 2/π ). Tables of the standard normal distribution are not yet available, and his calculations proceed
via the error function (see Stigler, 1986, pages 246–
248, e.g.), and prove to be remarkably accurate.
In previous work Fechner had introduced other
“main values” of a frequency distribution, the Zentralwert or “central value” C, and the Dichteste Wert or
“densest value” D, subsequently known in English as
the median and the mode. Arguing that the equality
of A, C and D is the exception rather than the rule, he
next introduces the Zweispaltiges Gauss’sche Gesetz to
represent this asymmetry. Calculating mean absolute
deviations from the mode separately for positive and
negative deviations from D, the “law of proportions”
is invoked, that these should be in the same ratio as
the numbers of observations on which they are based.
On converting from relative frequencies of observations to probabilities, and from subset mean absolute
deviations to subset standard deviations, it is seen that
this is exactly the requirement discussed above, that the
probabilities below and above the mode are in the ratio
σ1 /σ2 , to give a curve that is continuous at the mode.
Fechner says that he first discovered this law empirically, and warns that determination of the mode from
raw data is not straightforward. He goes on to show
that, in this distribution, the median lies between the
mean and the mode.
The first mathematical expression of the two normal
curves with different precision soon appears in what is
the first square-bracketed paragraph in the book and
the only such paragraph in Chapter 5. More extensive workings by Lipps appear in Chapter 19, “The
asymmetry laws,” where every paragraph is enclosed
in square brackets. Here Lipps traces the development
and properties of the distribution more formally, including an expression for the density function [equation (6), page 297] which corresponds to equation (1)
on converting between measures of dispersion. Nevertheless, the key steps in that development, in Chapter 5,
were Fechner’s alone.
We note that the second “generalisation” presented
later in Chapter 5 of Kollektivmasslehre is a form of
log-normal distribution, but this receives less emphasis
and is not our present focus of attention.
3. AN EARLY REDISCOVERY: EDGEWORTH
In 1898–1900 Edgeworth contributed a five-part article “On the representation of statistics by mathematical
formulae” to the Journal of the Royal Statistical Society, each part appearing in a different issue of the journal. His objective was “to recommend formulae which
have some affinity to the normal law of error, as being specially suited to represent statistics of frequency.”
The first two parts deal with the “method of translation,” or transformations to normality, and the “method
of separation,” or mixtures of normals, using modern
terminology.
In the third part Edgeworth considers the “method
of composition,” in which he constructs “a composite probability-curve, consisting of two half-probability
THE TWO-PIECE NORMAL DISTRIBUTION
curves of different types, tacked together at the mode,
or greatest ordinate, of each, so as to form a continuous whole, as in the accompanying figure” (1899,
page 373, emphasis in original; the figure is very similar to the solid line in Figure 1 above). He gives expressions for the two appropriately scaled half-normal
√
curves, as above, using the modulus, equal to 2 standard deviation, as his preferred measure of spread. He
says that this idea of two probability curves with different moduli is suggested by Ludwig (1898); however, its
development in the context of the normal distribution is
Edgeworth’s alone, since Ludwig’s comment comes in
a discussion of frequency curves based on the binomial
distribution.
To “determine the constants,” that is, estimate the parameters, given a sample mean and second and third
sample moments, Edgeworth rearranges their definitions to obtain a cubic equation in the distance between
the mean and the mode; the required parameter estimates follow from the real solution to this equation.
He gives a practical example and compares the method
of composition to the methods discussed earlier. In his
opinion, the “essential attribute” of the new method is
its “deficiency of a priori justification,” in contrast to
the normal distribution itself.
4. THE CRITICS: PEARSON AND SHEYNIN
The first English-language discussion of Fechner’s
contribution appears in a 44-page article by Karl Pearson, published in 1905 in Biometrika, the journal he
had co-founded four years earlier. The article is a response to a review of Pearson’s and Fechner’s works
on skew variation by Ranke and Greiner (1904) in the
leading German anthropology journal. Pearson’s title
quotes most of the title of the German article, omitting
its reference to anthropology, and adds the words “A rejoinder,” although the running head throughout his article is “Skew variation, a rejoinder.” He explains that
the German journal had provisionally accepted a rejoinder, but when it arrived the editors did not “see fit
to publish” his reply, so he placed it in Biometrika, of
which he was, in effect, managing editor. From a statistical point of view this seems to have been a more
appropriate outcome, since his article contains much
general statistical discussion and is most often cited for
its introduction of the terms platykurtic, leptokurtic and
mesokurtic.
However, Pearson’s article also contains extensive
attacks on Ranke and Greiner, who had argued that,
109
for the anthropologist, only the Gaussian law is of importance. In this respect the article is a good example of his well-documented behaviour. For example,
Stigler (1999, Chapter 1) opens by observing that “Karl
Pearson’s long life was punctuated by controversies,
controversies he often instigated, usually pursued with
a zealous energy bordering on obsession;” he “was a
fighter who vigorously reacted against opinions that
seemed to detract from his own theories. Instead of
giving room for other methods and seeking cooperation, his aggressive style led to controversy” (Hald,
1998, page 651); he was ever “relentless in controversy” (Cox, 2001, page 5) and “beyond question a
fierce antagonist” (Porter, 2004, page 266). Some of
this antagonism is directed towards Fechner: although
Pearson and Fechner are on the same side of the debate with Ranke and Greiner about asymmetry, Pearson sees “Fechner’s double Gaussian curve” as a rival
to his family of curves, and criticises it on both statistical and historical grounds.
Using the parameterisation in terms of σ1 and σ2 as
in equation (1), Pearson presents expressions for the
first four moments of the distribution. Rather than “the
rough process by which Fechner determines the mode
and obtains the constants of the distribution,” he shows
that “fitting by my method of moments is perfectly
straightforward.” To do this, he obtains the cubic equation discussed above, and says in a footnote (page 197)
“This cubic was, I believe, first given by Edgeworth,”
but there is no reference. He observes that the skewness and kurtosis are not independent of one another,
so that “we cannot have any form of symmetry but
the mesokurtic.” He obtains the bounds on β2 given
above, but notes that many empirical distributions with
values outside this range have been observed. Hence,
Pearson’s overall conclusion is that “the double Gaussian curve fails us hopelessly.” Curiously, having defined platykurtic as “more flat-topped” and leptokurtic
as “less flat-topped” than the normal curve, as has become standard usage, he contrarily describes Fechner’s
double Gaussian curve as platykurtic, despite having
shown its positive excess kurtosis. Similarly, another
curve, the symmetrical binomial, is said to be “essentially leptokurtic, that is, β2 < 3” (page 175).
Turning to questions of precedence, Pearson’s
counter claims appear in a footnote (page 196) at the
start of the statistical discussion summarised above,
which reads as follows:
Here again it is historically incorrect to attribute these curves to Fechner. They had
110
K. F. WALLIS
been proposed by De Vries in 1894, and
termed “half-Galton curves,” and Galton
was certainly using them in 1897. See the
discussion in Yule’s memoir, R. Statist. Soc.
Jour. Vol. LX, page 45 et seq.
Pearson was familiar with De Vries (1894), having
used two of his J-shaped botanical frequency distributions as Examples XI and XII in his 1895 article
on skew variation. De Vries said that these deserved
the name half-Galton (i.e., half-normal) simply on the
basis of the appearance of the empirical distributions,
and no fitting was attempted, nor did he make any proposal to place two such curves together to give a more
general asymmetric distribution. Fechner’s curve had
not been proposed by De Vries. [Edgeworth knew that
his composite curve had not either, noting at the outset
(1899, page 373) that “It will be observed that the following construction is not much indebted to the “halfGaltonian” curve employed by Professor De Vries.”]
Galton comes a little closer, but Pearson is again incorrect. His citation is inaccurate, since he clearly has
in mind Yule’s paper read at the Royal Statistical Society in January 1896, published with discussion later
that year (Yule, 1896a). Galton had opened the discussion at the meeting and mentioned his method of percentiles as an alternative to the method of frequency
curves developed by Pearson and applied by Yule. In
response to a request at the meeting, he provided a
memorandum giving fuller information on his method,
which was published in the same issue of the Society’s journal (Galton, 1896), together with a reply by
Yule (1896b). Galton explains how his method of percentiles, in this example method of deciles, smooths
the original frequency table or “frequency polygon”
of Yule by interpolating deciles and plotting them. He
then mentions another approach, namely
. . . the extremely rude and scarcely defensible method, but still a sometimes serviceable one, of looking upon skew-curves as
made up of the halves of two different normal curves pieced together at the mode. . . .
On trying it, again for curiosity’s sake, with
the present series for all the five years, there
was of course no error for the 2nd, 5th, and
8th deciles, . . .
because he had inferred the spread or standard deviation of the lower half-normal distribution from the
lower 20% point of the standard normal distribution,
and similarly for the upper part; he goes on to discuss the errors of fit at the other deciles. But no “law
of proportions” or scaling is applied, and the resulting curve is discontinuous, like the initial two halves of
normal curves in Figure 1. Yule (1896b) recognises this
in his response, noting that, in contrast, his skew-curve
“presents a continuous distribution round the mode.”
Galton was certainly not using Fechner’s curves.
The erroneous assertions in Pearson’s footnote may
be due to his combativeness. Several authors also discuss the tremendous volume of work he undertook. For
example, Cox (2001, page 6) observes that he “wrote
more than 90 papers in Biometrika in the period up
to 1915, few of them brief, and appears to have been
the moving spirit behind many more.” He founded not
only the journal but also the Biometrics Laboratory
at University College London at this time. His son
Egon remarks that the volume of work “led inevitably
to a certain hurry in execution” (E. S. Pearson, 1936,
page 222). This remark is made during discussion of
one of Pearson’s two well-known errors, recently reappraised by Stigler (2008), but it perhaps also applies to
the mistakes discussed above, which are of a smaller
order of magnitude. Nevertheless, Pearson’s assertions
in the quoted footnote are mistaken, and his challenge
to Fechner’s claim to priority is unjustified, and thereby
unjust.
Sheynin (2004), in his review of Fechner’s statistical work, has a very brief discussion of the doublesided Gaussian law, quoting from sections of Kollektivmasslehre that had been worked on by Lipps, and
hence underestimating the role of Fechner’s law of proportions. In his discussion (page 68) he states that the
double-sided Gaussian law was not original to Fechner,
this having been pointed out, forcefully, by Pearson
(1905). As if quoting from Pearson, and giving no citation for De Vries (1894), Sheynin states “De Vries,
in 1894, had applied the double-sided law.” In this
statement “applied” is somewhat stronger than Pearson’s “proposed,” hence is further from the truth, and
Sheynin’s denial of Fechner’s originality is similarly
inaccurate and unjust.
5. LATER REDISCOVERIES AND EXTENSIONS
The result of Pearson’s critique appears to have been
that, with two exceptions discussed below, the Fechner distribution, with this attribution, disappeared from
the statistical literature until its reappearance in Hald’s
(1998) history. Meanwhile, three independent rediscoveries occurred.
First, in the physics literature, is Gibbons and Mylroie’s (1973) “joined half-Gaussian” distribution, cited
111
THE TWO-PIECE NORMAL DISTRIBUTION
by Johnson, Kotz and Balakrishnan (1994), as noted
above; the distribution is fitted by what statisticians
recognise as the method of moments. Second, in the
statistics literature, is the “three-parameter two-piece
normal” distribution of John (1982), also cited by Johnson, Kotz and Balakrishnan (1994); John compares estimation by the method of moments and maximum
likelihood. In the same journal Kimber (1985) notes
that John (1982) is a rediscovery, with reference to
Gibbons and Mylroie (1973); he proves the asymptotic
normality of ML estimators and provides a likelihood
ratio test of symmetry. Finally, in the meteorology literature, Toth and Szentimrey (1990) introduce the “binormal” distribution, again fitted by ML, with a test
of symmetry. The same name is used by Garvin and
McClean (1997), who nevertheless again attribute the
distribution to Gibbons and Mylroie. In all these articles the distribution is parameterised in terms of the
mode, using various symbols, and the standard deviations σ1 and σ2 , as in (1) above. An alternative parameterisation, with a single explicit skewness parameter,
is given by Mudholkar and Hutson (2000), who do acknowledge Fechner’s priority.
A modern, but pre-Hald (1998) attribution to Kollektivmasslehre occurs at the start of an exploration by
Runnenburg (1978) of the mean, median, mode ordering. He notes that Fechner had shown this Lagegesetz
der Mittelwerte for the two-piece normal distribution,
and investigates more general conditions in which it
holds. The second exceptional appearance of the Fechner distribution in the statistical literature pre-1998 is
more substantial. Barnard (1989), seeking a family of
distributions “which may be expected to represent most
of the types of skewness liable to arise in practice,” introduces the distribution
⎧
1 −M(x − μ) a
⎪
⎪
⎪
K
exp
−
,
x ≤ μ,
⎨
2
σ
f (x) =
⎪
1 x −μ a
⎪
⎪ K exp −
,
x ≥ μ,
⎩
2
σ
which reparameterises and generalises the two-piece
normal distribution in equation (1). He calls it the
Fechner family, because by allowing the skewness parameter M (M > 0) to differ from 1 it embodies Fechner’s idea in Kollektivmasslehre of having different
scales for positive and negative deviations from the
mode, μ. It also allows for nonnormal kurtosis by allowing a (1 ≤ a < ∞) to differ from 2. The scale parameter σ is equal to the standard deviation if (M, a) =
(1, 2) but not otherwise, in general. This “Fechner family of unimodal densities” also appears in a later article (Barnard, 1995), which is cited by Hald (1998,
page 380). We note that the case a = 1, the asymmetric Laplace distribution, has a considerable life of
its own, beginning before Barnard’s work: see, for example, Kotz, Kozubowski and Podgorski (2001, Chapter 3) and the references therein.
Two further extensions of note, independent of
Fechner, can be found in Bayesian statistics. For the
application of Monte Carlo integration with importance sampling to Bayesian inference, Geweke (1989)
uses “split” (i.e., two-piece) multivariate normal and
Student-t distributions as importance sampling densities. The generalisation by Fernandez and Steel (1998)
is also cast in a Bayesian setting: as in Barnard’s Fechner family, there is a single skewness parameter, which
is convenient whenever it is desired to assign priors
to skewness; nevertheless, it has general applicability.
For any univariate PDF f (x) which is unimodal and
symmetric around 0, Fernandez and Steel’s class of
two-piece or split distributions p(x|γ ), indexed by a
skewness parameter γ (γ > 0), is
(4)
p(x|γ ) =
⎧
(γ x),
⎨ Kf x ≤ 0,
⎩ Kf
x ≥ 0,
x
,
γ
−1
where K = 2 γ + γ −1 . If γ > 1 there is positive skewness, and inverting γ produces the mirror image of the density function around 0. Unlike Barnard’s
Fechner family there is no explicit kurtosis parameter; kurtosis is introduced, if desired, by the choice of
f (x), most commonly as Student-t. An extension with
two tail parameters to allow different tail behaviour in
an asymmetric two-piece t-distribution is developed by
Zhu and Galbraith (2010).
ACKNOWLEDGMENTS
I am grateful to Sascha Becker, David Cox, Chris
Jones, Malte Knüppel, Kevin McConway, Mary Morgan, Oscar Sheynin, Mark Steel, Stephen Stigler and
two referees for comments and suggestions at various stages of this work. I am especially indebted to
Karl-Heinz Tödter for his expert assistance with the
German-language works cited. Thanks also to the Resource Delivery team at the University of Warwick Library.
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